Keywords: Polarizability, two-level atoms

Keywords: Polarizability, two-level atoms

The experiments here described have arisen from preliminary work conducted with the ultimate object of testing the late Prof. Callendar's formula* for the distribution of energy in the spectrum. The latter experiments, not yet completed, involved radiation-pressure determinations made with a vane suspended in a vacuum sufficiently high for radiometer action to be completely eliminated. While the manner in which radiometer action diminishes as the vacuum improves was being investigated, the series of radiometer effects recorded in the first part of this paper was observed. Radiation was directed upon a light vane suspended by means of a fine quartz fibre in a flask, and the deflections produced when the radiation was incident in turn upon the two sides of the vane were observed for a series of vacuum conditions. The vanes employed included platinized or silvered glass and mica, and also an aluminium vane. The range of air pressure used extended down to about 10^-6 mm. of mercury.

In the case of the platinized glass vane, at a pressure of a few mm. of mercury the deflections on both the glass and the platinized sides were of the order of those registered at the highest vacua obtained. At pressures of about 10-2 mm. gas effects developed which were very large in the case of the glass and mica vanes but much less marked with the aluminium vane. With all the vanes, gas action had practically vanished at a pressure of 10-6 mm. of mercury. Under these conditions, measurements of radiation pressure, using glass and mica vanes, were found to give results in agreement with the energy-density of the radiation to within ± 7 per cent.

While the above preliminary experiments were in progress the errors of calculation occurring in Nichols and Hull's paper dagger on radiation pressure were noticed. Although their work, carried out at a gas pressure of 16 mm. of mercury, has been widely quoted as conclusively establishing, to within about 1 per cent, the numerical equivalence between the pressure and energy-density of radiation, it is found that their results, when correctly evaluated, show a divergence between these quantities of some 10 per cent. Hence Nichols and Hull's investigation cannot be regarded as furnishing a quantitative experimental verification of the equality relationship deducible from theory.

It was therefore thought worth while, using the working experience gained, to attempt more accurate measurements of radiation pressure by the direct method of using metal vanes and a vacuum so high that radiometer action was eliminated. In the second part of the paper the observations and results for all the radiation-pressure measurements are given. With each vane a considerable range of radiation-intensities was employed, the maximum deflections of the suspensions under the influence of the radiation-pressure being up to 10 or 15 times those obtained by Nichols and Hull. The corresponding energy-densities of the radiation were measured by the use of a Callendar cup radio-balance. The results for an alloy vane, probably the most reliable of those derived, show that for a number of radiation-intensities the difference between pressure and energy-density varies from +4 to -3 per cent. The present experiments are considered in the light of those conducted by other observers. Errors of calculation arising in Nichols and Hull's results are examined in the appendix

The scanning tunneling miscroscope is proposed as a method to measure forces as small as 10^-18N. As one application of this concept, we introduce a new type of miscroscope capable of investigating surfaces of insulators on the atomic scale. The atomic force miscroscope is a combination of the principles of the scanning tunneling microscope and the stylus profilometer. It incorporates a probe that does not damage the surface. Our preliminary results in air demonstrate a lateral resolution of 30 Å and a vertical resolution less than 1 Å.

Elementary considerations support the result of Abraham, by which the momentum is, for a given energy, inversely proportional to the refractive index, rather than proportional it, as predicted by Minkowski. This conclusion disagrees, however, with the experiment of Jones and Richards

The controversial issue of whether Minkowski's tensor or Abraham's tensor is the appropriate electromagnetic stress-energy-momentum tensor form in ponderable media is investigated. Various aspects of differences between the two tensors are discussed. As far as experiments are concerned, the predictions of Minkowski's tensor are consistent with experiments, but Abraham's tensor fails in some experiments. The problem of reflection and transmission by a moving dielectric medium offers a good example for illustrating the applications of the electromagnetic stress-energy-momentum tensor. It is shown by the use of the stationary phase method that Minkowski's tensor is the appropriate electromagnetic stress-energy-momentum tensor

Quantum zeta function Z(s)= E^-s is defined as a sum over energy eigenvalues E. It has exact solutions in cases where E is not known.

A general introduction surveying the problems to be examined in a series of papers is followed by a detailed treatment of the magnetic behaviour of a large system of electrons. The Schrödinger equation is solved on the assumption that the system is unbounded, and the modifications caused by the finite size of the system are then determined for the limiting case in which the system is much larger than the electronic orbits. An expression is then obtained for the density of states, and the free energy of the system found assuming that kT<E_o, where E_o is the degeneracy parameter. The magnetic susceptibility, thermodynamic potential and specific heat are discussed for the two cases N constant and E_o constant. Explicit formulae are given for the temperature-dependence of the field-independent term in the susceptibility. In the final section the corrections due to the electron spin are introduced.

A discussion of the effects of collisions on the magnetic properties of a large system of free electrons shows that the non-periodic term in the susceptibility is hardly affected, but that the periodic terms are reduced in magnitude by a factor exp(-hp/taubeta H), where p is the harmonic considered, tau is the mean collision time, and beta=eh/2pi mc.

It is shown that electromagnetic radiation incident on a large system of electrons moving in a constant magnetic field H in a metal is strongly absorbed near a frequency nu=eH/2pi mc, where m is the effective mass. The resonance absorption is found to be of the same order of magnitude as the absorption due to skin effect.

The first part of the paper consists of a calculation of the magnetic properties of a system of electrons within a cylinder of very small radius placed with its axis parallel to the field direction. Perturbation theory is used to find the energy levels, and from these the number of occupied states in a two-dimensional system is determined by summation over the quantum numbers; the results are then generalised to the three-dimensional case. The expressions for magnetic susceptibility, thermodynamic potential, and specific heat are found to contain a steady term which remains of significant magnitude at all temperatures, together with terms periodic in the field which are significant only at very low temperatures. The influence of electron spin on both steady and periodic terms is discussed. The second part consists of similar calculations for a small spherical system.

The WKB method is used to solve the Schrödinger equation for an electron moving in a uniform magnetic field H, the boundary of the system being a cylinder with its axis lying along the direction of the field. It is found that there are two entirely different types of wavefunction possible, one type leading to small Landau diamagnetism of large systems discussed in Part 1 of this series, the other to the larger diamagnetism of small systems discussed in Part 4. Taking into account the occupied states of both types, the steady (non-periodic) contributions to the magnetic susceptibility are derived for all fields in both the low-and high-temperature limits, and for most fields at intermediate temperatures.

This paper considers the evolution of the relation between theory and cosmology from the development of the first simple quantitative models in 1917 to the sophistication of our cosmological models at the turn of the millenium.

When a dielectric object is placed between two opposed, nonfocused laser beams, the total force acting on the object is zero but the surface forces are additive, thus leading to a stretching of the object along the axis of the beams. Using this principle, we have constructed a device, called an optical stretcher, that can be used to measure the viscoelastic properties of dielectric materials, including biologic materials such as cells, with the sensitivity necessary to distinguish even between different individual cytoskeletal phenotypes. We have successfully used the optical stretcher to deform human erythrocytes and mouse fibroblasts. In the optical stretcher, no focusing is required, thus radiation damage is minimized and the surface forces are not limited by the light power. The magnitude of the deforming forces in the optical stretcher thus bridges the gap between optical tweezers and atomic force microscopy for the study of biologic materials.

The theory of quantum-mechanical grand canonical ensembles is used to derive for the case of a perfect Bose-Einstein gas the average number of particles in the different energy levels, the fluctuations in these numbers and the equation of state. The Einstein condensation phenomenon is then discussed, and it is shown that in a p-v diagram (v being the specific volume) the isotherm consists of two analytically different parts in the limit where the number of particle in the system, N, goes to infinity. It is also shown that for finite N at the critical volume ⁿp/ vⁿ is of the order of N^(n-2)/2 in accordance with a result obtained by Wergeland and Hove-Storhoug

Keywords: Angular momentum, geometrical gauge

SUMMARY ONLY: Using adiabatic invariance, energy conservation and energy quantization the radiation force f due to a reversible or weakly attentuated beam of energy in an isotropic medium is shown to be f=W/c=Ev_g/v_p where W is the beam's power, E is its energy per unit lenght, v_p is phase velocity and v_g its energy velocity = group velocity = particle velocity.

The force on a Clausius-Mossotti fluid is an electromagnetic field is evaluated microscopically. Owing to the modification of the two-particle density by the electric field, an additional mechanical force Delta f ^M is found. When added to the electrical force F ^E, the total force in the static case becomes identical to that deduced microscopically by Helmholtz. The analysis is extended to various time-dependent cases, and it is pointed out that Delta f ^M essentially assumes its static value on time scales longer that T _c, the relaxation time of the two-particle density, but is otherwise negligibly small. This Peierls's theory of the momentum of light is valid only for pulses shorter than T _c; the necessary correction due to Delta f ^M in other cases is given and discussed.

The Abraham force is discussed briefly. It is recalled that the old controversy concerning the Minkowski and Abraham tensors for an electromagnetic field in matter, as well efforts at observing the Abraham force, are pointless because it is now well established that both tensors are equally satisfactory, as long as the various subsystems are treated correctly. If one ascribes to the electromagnetic field in matter a momentum density of E× H/c², like Abraham, then Lahoz and Graham have observed the magnetic force density on a polarization current, or only one of six components of the Abraham force. Their interpretation of their experiments is erroneous.

The classical theory of the electromagnetic field associated with paraxial Laguerre-Gaussian light is generalized to obtain expressions for the electric and magnetic field operators. The forms of the Poynting vector and angular momentum density operators are derived and their expectation values for a single-photon wave packet are obtained. The Lorentz force operator in the dielectric is resolved into longitudinal, radial and azimuthal components. The theory is extended to apply to an interface between two semi-infinite dielectric media, one of which is transparent with an incident single-photon pulse, and the other of which is weakly attenuating. For a pulse that is much shorter than the attenuation length, the theory can separately identify the surface and bulk contributions to the Lorentz force on the attenuating dielectric. Particular attention is given to the transfer of longitudinal and angular momentum to the dielectric from light incident from free space. The resulting expressions for the shift and rotation of a transparent dielectric slab are shown to agree with those obtained from Einstein box theories.

The momentum transfer from light to a dielectric material in the photon drag effect is calculated by evaluation of the relevant Lorentz force. In accordance with measurements on Si and Ge, the material is taken as a two-component optical system, with charge carriers described by an extinction coefficient kappa in a host semiconductor described by real refractive indices eta _p (phase) and eta _g (group). The calculated momentum transfer to the charge carriers alone has the value eta _pomega/c per photon found experimentally. The time-dependent Lorentz force is calculated for light in the form of a narrow-band single-photon pulse. When the pulse is much shorter than the attenuation length, which is much shorter than the sample thickness, there is a clear separation in time between surface and bulk contributions to the forces. The total bulk momentum transfer (charges plus host) in this case is found to be omega/eta _gc.

Keywords: radiation pressure; dielectric materials; extinction coefficients; momentum; refractive index

Keywords: Polarizability, two-level atoms

Laser beams exert torque on microparticles through very different physical mechanisms. In this paper, optical angular momentum transferred by laser light to a trapped absorbing superparamagnetic microsphere has been studied, distinguishing between different contributions. We have found the main contribution to the torque arising from the transfer of the spin angular momentum carried by absorbed laser light. Detailed polarization status contribution of the laser light to the momentum transfer has been then analyzed. A general method to separate and quantify contributions to the optical angular momentum transferred has been developed. We have thus quantified contributions due to radiation pressure, through an effect similar to the wind on a windmill, and contributions arising from magneto-optic effects.

The properties of ultrashort gaussian pulse propagation in a dispersive, attenuative medium are reviewed with emphasis on the pulse velocity. Of particular interest is the group velocity whose physical interpretation loses meaning in causally dispersive materials as the temporal pulse width decreases into the ultrashort pulse regime. A generalized definition of the group velocity that applies to ultrashort pulses in causally dispersive materials is provided by the centroid velocity of the pulse Poynting vector whose properties are described here. In particular, it is shown that this physical velocity measure approaches the group velocity for any value of the initial pulse carrier frequency and at any fixed value of the propagation distance in the limit as the initial pulse width increases indefinitely. This then provides a convenient measure for determining when the group velocity approximation is valid.

Keywords: Polarizability, two-level atoms

Keywords: Polarizability, two-level atoms

In classical theory, a physical state is an equivalence class under gauge transformations. Is the same true in quantum theory? The physical quantum states are the solutions of Dirac’s quantum constraint equation. They cannot be constructed as equivalence classes under the simple gauge transformations generated by the Dirac constraints. However, we show here that they can be constructed as equivalence classes under suitably defined complete gauge transformations. The complete gauge transformations are generated by the action of the quantum constraints on arbitrary individual components of the state.

By means of the Helmholtz theorem on the decomposition of vector fields, the angular momentum of the classical electromagnetic field is decomposed, in a general and manifestly gauge invariant manner, into a spin component and an orbital component. The method is applied to linearly and circularly polarized plane waves in their classical and quantum forms.

Accurate measurements of the Abraham density force have been conducted in a BaTiO₃ ceramic for the first time. The measurement torque amplitudes (in the order of 10^-10 Nm) were in good agreement with the predicted values inside the error limits of 10%. This conclusively rules out Minkowski's unsymmetric momentum-energy tensor for material media at low frequencies.

A general approach to the description of an angle and phase is given on the basis of the kq representation. It is shown that an angular coordinate in quantum mechanics has to be treated as a quasicoordinate in order to avoid inconsistencies. The kq representation leads to a consistent definition of the angular-momentum-angle degree of freedom. Using the correspondence between classical and quantum mechanics for the phase of a harmonic oscillator, operators are defined that form a new quantum-mechanical representation. This representation clarifies the concept of the phase and sheds light on the general understanding of rotations in quantum mechanics.